Li-Ping Yang

Coarse-graining renormalization by higher-order singular value decomposition

We propose a novel coarse-graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two or three dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the three-dimensional Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.

Graphene on incommensurate substrates: Trigonal warping and emerging Dirac cone replicas with halved group velocity

The adhesion of graphene on slightly lattice-mismatched surfaces, for instance, of hexagonal boron nitride (hBN) or Ir(111), gives rise to a complex landscape of sublattice symmetry-breaking potentials for the Dirac fermions. Whereas a gap at the Dirac point opens for perfectly lattice-matched graphene on hBN, we show that the small lattice incommensurability prevents the opening of this gap and rather leads to a renormalized Dirac dispersion with a trigonal warping. This warping breaks the effective time-reversal symmetry in a single valley. On top of this an additional set of massless Dirac fermions is generated, which is characterized by a group velocity that is about half the one of pristine graphene.

Rigorous solution of the spin-1 quantum ising model with single-ion anisotropy

We solve the spin-1 quantum Ising model with single-ion anisotropy by mapping it onto a series of segmented spin-1/2 transverse Ising chains, separated by the S(z)=0 states called holes. A recursion formula is derived for the partition function to simplify the summation of hole configurations. This allows the thermodynamic quantities of this model to be rigorously determined in the thermodynamic limit. The low temperature behavior is governed by the interplay between the hole excitations and the fermionic excitations within each spin-1/2 Ising segment. The quantum critical fluctuations around the Ising critical point of the transverse Ising model are strongly suppressed by the hole excitations.

Exact solutions of a class of S = 1 quantum Ising spin models

We propose a hole decomposition scheme to exactly solve a class of spin-1 quantum Ising models with transverse or longitudinal single-ion anisotropy. In this scheme, the spin-1 model is mapped onto a family of the S=1/2 transverse Ising models, characterized by the total number of holes. A recursion formula is derived for the partition function based on the reduced S=1/2 Ising model. This simplifies greatly the summation over all the hole configurations. It allows the thermodynamic quantities to be rigorously determined in the thermodynamic limit. The ground-state phase diagram is determined for both the uniform and dimerized spin chains. The corresponding thermodynamic properties are calculated and discussed.

Progress towards quantum simulating the classical O(2) model

Department of Energy under DOE [DE-FG02-05ER41368, DE-SC0010114, DE-FG02-91ER40664]; Army Research Office of the Department of Defense [W911NF-13-1-0119]; NSF [DMR-1411345]

Fine structure of the entanglement entropy in the O(2) model

We compare two calculations of the particle density in the superfluid phase of the O(2) model with a chemical potential μ in 1+1 dimensions. The first relies on exact blocking formulas from the Tensor Renormalization Group (TRG) formulation of the transfer matrix. The second is a worm algorithm. We show that the particle number distributions obtained with the two methods agree well. We use the TRG method to calculate the thermal entropy and the entanglement entropy. We describe the particle density, the two entropies and the topology of the world lines as we increase μ to go across the superfluid phase between the first two Mott insulating phases. For a sufficiently large temporal size, this process reveals an interesting fine structure: the average particle number and the winding number of most of the world lines in the Euclidean time direction increase by one unit at a time. At each step, the thermal entropy develops a peak and the entanglement entropy increases until we reach half-filling and then decreases in a way that approximately mirrors the ascent. This suggests an approximate fermionic picture.

Estimating the central charge from the Rényi entanglement entropy

Author(s): Bazavov, A; Meurice, Y; Tsai, SW; Unmuth-Yockey, J; Yang, LP; Zhang, J | Abstract:

Probing the conformal Calabrese-Cardy scaling with cold atoms

We demonstrate that current experiments using cold bosonic atoms trapped in one-dimensional optical lattices and designed to measure the second-order Renyi entanglement entropy S_2, can be used to verify detailed predictions of conformal field theory (CFT) and estimate the central charge c. We discuss the adiabatic preparation of the ground state at half-filling where we expect a CFT with c=1. This can be accomplished with a very small hoping parameter J, in contrast to existing studies with density one where a much larger J is needed. We provide two complementary methods to estimate and subtract the classical entropy generated by the experimental preparation and imaging processes. We compare numerical calculations for the classical O(2) model with a chemical potential on a 1+1 dimensional lattice, and the quantum Bose-Hubbard Hamiltonian implemented in the experiments. S_2 is very similar for the two models and follows closely the Calabrese-Cardy scaling, (c/8)\ln(N_s), for N_s sites with open boundary conditions, provided that the large subleading corrections are taken into account.

Tensor Renormalization Group Study of the General Spin-S Blume–Capel Model

We focus on the special situation of

A quantum transfer matrix method for one-dimensional disordered electronic systems

D=2J